Mathematical Series
Mathematical series representations are very useful tools for describing images or for solving/approximating
the solutions to imaging problems. The may be used to “expand” a function into terms that are individual
monomial expressions (i.e., “powers”) of the coordinate
Geometric Series
Adjacent terms in a geometric series exhibit a constant ratio, e.g., if the scale factor for adjacent terms in
the series is t, the series has the form:
1 + t + t2 + t3 + · · · =
∞
X
tn
n=0
If |t| < 1, this solution converges to a simple (and EASILY remembered) expression:
∞
X
tn =
n=0
1
if |t| < 1
1−t
This series pops up frequently in science and it is useful to remember the solution. We may “turn the
problem around” by using a truncated series as an approximation for the ratio:
∞
N
X
X
1
tn ∼
tn
=
=
1 − t n=0
n=0
where N is some maximum power in the series
Examples:
1.
1
0.9
1
0
1
2
3
= (0.1) + (0.1) + (0.1) + (0.1) + · · ·
1 − 0.1
= 1 + 0.1 + 0.01 + 0.001 + · · · = 1.11111 · · ·
=
This series converges fairly quickly and because of the small value of t; this series may be truncated
after a few terms and still obtain a fairly accurate value.
2.
1
0.25
= 4
1
= (0.75)0 + (0.75)1 + (0.75)2 + (0.75)3 + (0.75)4 + (0.75)5 + · · ·
1 − 0.75
= (1 + 0.75 + 0.5625 + 0.421875 + 0.31640635) + · · ·
= 3.05078125 + · · · < 4
=
Note that this series converges slowly because t is relatively “large;” the sum of a few terms is a poor
approximation to the end result.
1
Finite Geometric Series
The truncated geometric series also may be rewritten into a simple expression. Consider the finite series
that inclues N + 1 terms:
N
X
n=0
tn = 1 + t + t2 + t3 + · · · + tN
=
∞
X
n=0
∞
X
tn −
tn
n=N +1
We may write this as the difference of two infinite geometric series:
N
X
tn
=
n=0
=
¡
¢ ¡
¢
1 + t + t2 + t3 + · · · tN + tN +1 + · · · − tN +1 + tN+2 + · · ·
∞
X
n=0
n
t −
∞
X
tn
n=N+1
Now change the summation variable for the second infinite series from n to u ≡ n − (N + 1) =⇒ n =
u + N + 1:
∞
∞
∞
∞
X
X
X
X
1
tn =
tu+(N +1) =
tu tN +1 = tN+1 ·
tu = tN +1 ·
1
−
t
u=0
u=0
u=0
n=N +1
The expressions for the two series may now be combined:
N
X
tn
=
n=0
∞
X
n=0
=
µ
tn −
1
1−t
N
X
¶
∞
X
tn
n=N
tn =
n=0
− tN+1 ·
1
1−t
1 − tN +1
if |t| < 1
1−t
This is also often shown in the form where the maximum power in the series is N − 1 so that there are N
terms:
N−1
X
1 − tN
tn =
if |t| < 1
1−t
n=0
Examples:
1.
4
X
n
(0.1) = 1 + 0.1 + 0.01 + 0.001 + 0.0001 = 1.1111
n=0
t = 0.1, N = 4 =⇒
1 − (0.1)5
1 − tN +1
=
= 1.1111
1−t
1 − (0.1)
2.
3
X
(0.75)n = 2.7344
n=0
4
1 − (0.75)
1 − tN+1
=
= 2.7344
1−t
1 − (0.75)
2
Binomial Expansion:
1
n!
n
n (n − 1) 2 n (n − 1) (n − 2) 3
+ x+
x +
x + ··· +
xr + · · ·
0! 1!
2!
3!
(n − r)!r!
n (n − 1) 2 n (n − 1) (n − 2) 3
= 1 + nx +
x +
x + ···
¶
µ
¶
µ
¶ µ 2¶
µ
¶ 6µ
n
n
n
n
n
2
3
x + ··· +
xn−r xr + · · ·
≡
+
x+
x +
3
r
0
1
2
µ
¶
¶
∞ µ
X
n!
n
n
n
(1 + x) =
≡
· xn−r · xr where
and 0! ≡ 1
r
r
(n − r)!r!
r=0
(1 + x)n =
If n is a positive integer, the series includes n + 1 terms. If n is NOT a positive integer, the series converges
if |x| < 1. If n > 0, the series also converges if |x| = 1.
n (n − 1)
n (n − 1) (n − 2)
(−x)2 +
(−x)3 + · · ·
2
6
n (n − 1) 2 n (n − 1) (n − 2) 3
= 1 − nx +
x −
x + ···
2!
3!
Again, the series may be truncated to approximate the result. In many areas of science, the series is often
truncated after the second term so the summation only includes the constant and the linear term (e.g.,
Fresnel approximation to optical diffraction) so that:
(1 ± x)n ∼
= 1 ± nx if |x| ' 0
=⇒ (1 − x)n = 1 + n (−x) +
Examples:
1.
1
1−x
= (1 − x)
−1
= 1 − (−1) x +
(−1) (−2) 2 (−1) (−2) (−3) 3
x −
x + ···
2!
3!
2! 2 − (3!) 3
x −
x + ···
2!
3!
= 1 + x + x2 + x3 + · · ·
= 1+x+
which demonstrates that the geometric series may be written as a binomial expansion.
2. Square root
¡1¢ ¡ 1¢ ¡ 3¢
¡1¢ ¡ 1¢
p
−2
−2 −2
1
2
2
·x + 2
· x2 + · · ·
(1 + x) = (1 + x) = 1 + · x +
2
2
6
1
1
1
= 1 + · x − · x2 +
· x3 + · · ·
2
8
16
In words, this expresses the square root of 1 + x as a series of the powers of x with decreasing weights.
Truncation to two terms yields an approximation for the square root that is most accurate for |x| ' 0.
1
2
1
1
1
1
(1 + x) 2 ∼
= 1 + · x − · x2 ∼
=1+ ·x
2
8
2
17
1
= 1.0625 = 1 +
16
16
r
17
= 1.030776406 · · ·
16
sµ
¶
1 ∼
1+
first-order approximation:
=1+
16
sµ
¶
1 ∼
1+
second-order approximation:
=1+
16
3
1 1
33
·
=
= 1.03125
2 16
32
µ ¶2
2111
1 1
1 1
=
·
−
= 1.030761719 · · ·
2 16 8 16
2048
3. Cube root
¡1¢ ¡ 2¢ ¡ 5¢
¡1¢ ¡ 2¢
p
−3
−3 −3
1
2
3
3
· (−x) + 3
· (−x)3 + · · ·
(1 − x) = (1 − x) = 1 + · (−x) +
3
2
6
1
5
1
= 1 − · x − · x2 +
· x3 + · · ·
3
9
81
1
1
1
∼
= 1 + · x − · x2 ∼
=1+ ·x
3
9
3
1
3
1
15
=1−
16
16
µ ¶ 13
15
= 0.97871691 · · ·
16
µ
¶1
µ
¶
1 3 ∼
1
47
1
1−
=
= 0.979166666 · · ·
=1+ · −
16
3
16
48
¶1
µ
¶
µ ¶2
µ
2257
1
1
1
1
1 3 ∼
=
+ ·
= 0.979600694 · · ·
1−
=1+ · −
16
3
16
9
16
2304
µ
¶1
µ
¶
µ
¶2
µ
¶3
1 3 ∼
5
325 003
1
1
1
1
1
1−
+
=
+ · −
· −
= 0.979585624 · · ·
=1+ · −
16
3
16
9
16
81
16
331 776
4
Exponential Series
Without proof, we can write e raised to a numerical power u as a series in the powers of u:
exp [u] = eu =
∞
X
un
n!
n=0
1
u
u2 u3
+ +
+
+ ···
0! 1!
2!
3!
1
1
= 1 + u + u2 + u3 + · · ·
2
6
=
which may be easily generalized to any base:
au = exp [u · log [a]] =
e.g.
10u
∞
n
X
[u · log [a]]
n!
n=0
∞
X
[u · log [10]]n
n!
n=0
=
∞
X
(log [10])n n
·u
n!
n=0
∼
=
where log [10] ∼
= 2.302585
Complex Exponential Series
The generalization of the exponential series for complex-valued powers:
exp [±iθ] = e±iθ =
∞
n
X
(±iθ)
n!
n=0
1
(iθ) (iθ)2 (iθ)3 (iθ)4
±
+
∓
+
± ···
0!
1!
2!
3!
4!
1
1
1 5 5
1
= 1 ± iθ + i2 θ2 ∓ i3 θ3 + i4 θ4 ±
i θ + ···
2
6
24
120
µ
¶
µ
¶
θ2
θ4
θ3
θ5
= 1−
+
− ··· ± i · θ −
+
− ···
2
24
6
120
=
From Euler relation:
exp [+iθ] = cos [θ] + i sin [θ]
Equate real and imaginary parts:
θ2 θ4
θ2
θ4
1
−
+
− ··· = 1 −
+
− ···
0!
2!
4!
2
24
lim cos [θ] = 1
cos [θ] =
θ→0
sin [θ] = θ −
θ3 θ5
θ3
θ5
+
− ··· = θ −
+
− ···
3!
5!
6
120
lim sin [θ] = θ
θ→0
5
Maclaurin Series:
“Predict” the value of the function f [x] based on its value and its derivatives evaluated at the origin of
coordinates (x = 0):
¯
¯
¯
x2 d2 f ¯¯
x3 d3 f ¯¯
x0
x1 df ¯¯
+
+
+ ···
f [x] =
· f [0] +
·
·
·
0!
1! dx ¯x=0
2! dx2 ¯x=0
3! dx2 ¯x=0
¯
¶
∞ µ n
X
x
dn f ¯¯
=
·
n! dxn ¯x=0
n=0
¶
∞ µ n
X
x2 00
x3 000
x
0
(n)
= f [0] + x · f [0] +
· f [0] +
· f [0] + · · · =
· f [0]
2
6
n!
n=0
Taylor Series:
Generalization of the Maclaurin series to “predict” value of f [x + x0 ] based on the value of the function and
its derivatives evaluated at x0 :
¯
¯
¯
x00
x10 df ¯¯
x20 d2 f ¯¯
x30 d3 f ¯¯
+
+
+ ···
f [x + x0 ] =
· f [x]|x=x0 +
·
·
·
0!
1! dx ¯x=x0
2! dx2 ¯x=x0
3! dx2 ¯x=x0
¯
¯
¯
x0 df ¯¯
x20 d2 f ¯¯
x30 d3 f ¯¯
x0
+
+
+ ···
·
·
·
= 0 · f [x0 ] +
0!
1! dx ¯x=x0
2! dx2 ¯x=x0
3! dx2 ¯x=x0
Ã
!
¯
∞
X
xn0 dn f ¯¯
=
·
n! dxn ¯x=x0
n=0
¶
∞ µ n
X
x0
x20 00
x30 000
0
(n)
= f [x0 ] + x0 · f [x0 ] +
· f [x0 ] +
· f [x0 ] + · · · =
· f [x0 ]
2
6
n!
n=0
[Arfken, §1, Schey, Feynman V1 §2]
Could also think of it as evaluating the value of f [x] based on the function and its derivatives evaluated
at x − x0
¯
¯
¯
x0
x1 df ¯¯
x20 d2 f ¯¯
x30 d3 f ¯¯
f [x] = 0 · f [x0 ] + 0 ·
+
+
+ ···
·
·
0!
1! dx ¯x=x−x0
2! dx2 ¯x=x−x0
3! dx2 ¯x=x−x0
Ã
!
¯
∞
X
xn0 dn f ¯¯
=
·
n! dxn ¯x=x−x0
n=0
x2
x3
= f [x − x0 ] + x0 · f 0 [x − x0 ] + 0 · f 00 [x − x0 ] + 0 · f 000 [x − x0 ] + · · ·
2
6
¶
∞ µ n
X
x0
· f (n) [x − x0 ]
=
n!
n=0
Again, it often is useful to truncate the Taylor series to approximate the value of the function.
6